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Question 17901: A Finite Sum of Absolute Values Is Not an Infinite Certificate On this page Original question and givens Definition and scope Step-by-step derivation Reverse checks Result This question review and source binding Return to catalogue page 90
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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.
R0034-17901 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0051
Original question and givens Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary).
G001 — Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary).Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Step-by-step derivation S001 · implies Before
source operation=absseries-absolute-partial;original question snapshot retained;source presentation interface resolved in this page After
EXPLICIT ordered output fields=[terms, absolute_terms, absolute_partial_sum, boundary] Conditions: none
Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag.
Step identifiers and dependencies Rule: absseries_explicit_source_interface_fields_without_javascript · Uses: G001
S002 · implies Before
original ordered signed term index=0;value=1/2 After
absolute term at SAME index=1/2 Conditions: none
Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total.
Step identifiers and dependencies Rule: absseries_each_term_absolute_before_sum · Uses: G001, S001
S003 · calculation Before
finite absolute term1 sign reversal := mul(-1, -1/3) After
finite absolute term1 sign reversal = 1/3 Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies Before
original ordered signed term index=1;value=-1/3 After
absolute term at SAME index=1/3 Conditions: none
Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total.
Step identifiers and dependencies Rule: absseries_each_term_absolute_before_sum · Uses: G001, S003
S005 · calculation Before
finite absolute partial sum := sum(1/2, 1/3) After
finite absolute partial sum = 5/6 Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation Before
finite original signed sum for distinction := sum(1/2, -1/3) After
finite original signed sum for distinction = 1/6 Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies Before
finite signed terms=[1/2, -1/3];same-order absolute terms=[1/2, 1/3] After
sum absolute terms=5/6;absolute signed sum=1/6<=that;finite_absolute_partial_sum_only Conditions: none
These quantities may differ;both empty sums are0.This is the supplied finite data,not an absolute-convergence certificate for an unspecified infinite continuation.
Step identifiers and dependencies Rule: absseries_finite_absolute_sum_not_absolute_signed_sum_or_infinite_tail · Uses: G001, S006
Reverse checks S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute finite original signed sum for distinction with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute finite absolute partial sum with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute finite absolute term1 sign reversal with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.Result ([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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